This invention relates to estimating permeability.
Nuclear magnetic resonance (NMR) measurements typically are performed to investigate properties of a sample. For example, an NMR wireline or logging while drilling (LWD) downhole tool may be used to measure properties of subterranean formations. In this manner, the typical downhole NMR tool may, for example, provide a lithology-independent measurement of the porosity of a particular formation by determining the total amount of hydrogen present in fluids of the formation. Equally important, the NMR tool may also provide measurements that indicate the dynamic properties and environment of the fluids, as these factors may be related to petrophysically important parameters. For example, the NMR measurements may provide information that may be used to derive the permeability of the formation and viscosity of fluids contained within the pore space of the formation. It may be difficult or impossible to derive this information from other conventional logging arrangements. Thus, it is the capacity of the NMR tool to perform these measurements that makes it particularly attractive versus other types of downhole tools.
Typical NMR logging tools include a magnet that is used to polarize hydrogen nuclei (protons) in the formation and a transmitter coil, or antenna, that emits radio frequency (RF) pulses. A receiver antenna may measure the response (indicated by a received RF signal) of the polarized hydrogen to the transmitted pulses. Quite often, the transmitter and receiver antennae are combined into a single transmitter/receiver antenna.
The NMR techniques employed in current NMR tools typically involve some variant of a basic two step sequence that includes a polarization time and thereafter using an acquisition sequence. During the polarization time (often referred to as a xe2x80x9cwait timexe2x80x9d), the protons in the formation polarize in the direction of a static magnetic field (called B0) that is established by a permanent magnet (of the NMR tool). The growth of nuclear magnetization M(t) (i.e., the growth of the polarization) is characterized by the xe2x80x9clongitudinal relaxation timexe2x80x9d (called T1) of the fluid and its equilibrium value (called M0). When the specimen is subject to a constant field for a duration tp, the longitudinal magnetization is described by the following equation:                               M          ⁡                      (                          t              p                        )                          =                              M            0                    ⁡                      (                          1              -                              ⅇ                                                      -                    tp                                                        T                    1                                                                        )                                              (                  Eq          .                      xe2x80x83                    ⁢          1                )            
The duration of the polarization time may be specified by the operator (conducting the measurement) and includes the time between the end of one acquisition sequence and the beginning of the next. For a moving tool, the effective polarization time also depends on tool dimensions and logging speed.
Referring to FIG. 1, as an example, a sample (in the formation under investigation) may initially have a longitudinal magnetization 10 (called MZ) of approximately zero. The zero magnetization may be attributable to a preceding acquisition sequence, for example. However, in accordance with Equation (Eq.) 1, the MZ magnetization 10 (under the influence of the B0 field) increases to a magnetization level (called M(tp(1))) after a polarization time tp(1) after zero magnetization. As shown, after a longer polarization time tp(2) from zero magnetization, the MZ magnetization 10 increases to a higher M(tp(2)) magnetization level.
An acquisition sequence (the next step in the NMR measurement) typically begins after the polarization time. For example, an acquisition sequence may begin at time tp(1), a time at which the MZ magnetization 10 is at the M(tp(1)) level. At this time, RF pulses are transmitted from a transmitter antenna of the NMR tool. The pulses, in turn, produce spin echo signals 16 that appear as a RF signal to the NMR tool. A receiver antenna (that may be formed from the same coil as the transmitter antenna) receives the spin echo signals 16 and stores digital signals that indicate the spin echo signals 16. The initial amplitudes of the spin echo signals 16 indicate a point on the MZ magnetization 10 curve, such as the M(tp(1)) level, for example. Therefore, by conducting several measurements that have different polarization times, points on the MZ magnetization 10 curve may be derived, and thus, the T1 time for the particular formation may be determined.
As a more specific example, for the acquisition sequence, a typical logging tool may emit a pulse sequence based on the CPMG (Carr-Purcell-Meiboom-Gill) pulse train. The application of the CPMG pulse train includes first emitting a pulse that rotates the magnetization, initially polarized along the B0 field, by 90xc2x0 into a plane perpendicular to the B0 field. A train of equally spaced pulses, whose function is to maintain the magnetization polarized in the transverse plane, follows. In between the pulses, magnetization refocuses to form the spin echo signals 16 that may be measured using the same antenna. Because of thermal motion, individual hydrogen nuclei experience slightly different magnetic environments during the pulse sequence, a condition that results in an irreversible loss of magnetization and consequent decrease in successive echo amplitudes. This rate of loss of magnetization is characterized by a xe2x80x9ctransverse relaxation timexe2x80x9d (called T2) and is depicted by the decaying envelope 12 of FIG. 1. This may be referred to as a T2-based experiment.
The relaxation times may be used to estimate the permeability of a downhole formation. In this manner, the magnetic resonance relaxation-time of a water filled pore (of the formation) is proportional to a volume-to-surface ratio of the pore. A high surface-to-volume ratio indicates either the presence of clay minerals in the pore space or microporosity, both of which impede fluid flow. Therefore, there is a correlation between the magnetic resonance relaxation times and permeability.
Obtaining T2 times from magnetic resonance logs is an ill-posed problem. Either the precision or the resolution of the decay-time spectrum is severely limited by the signal to-noise ratio of the measurements. Quite often, magnetic resonance logs are depth-stacked before signal processing to improve the signal-to-noise ratio of the data. Depth stacking increases the signal-to-noise ratio (SNR) by adding, or stacking, the amplitudes of corresponding spin echo signals that are obtained from different NMR measurements. For example, the amplitude of the tenth spin echo signal from a first CPMG measurement may be combined with the amplitude of the tenth spin echo signal from a second CPMG measurement. Because the tool may be moving, the CPMG measurements are performed at different depths.
The above-described depth stacking increases the signal-to-noise ratio by a factor of {square root over (N)}, where xe2x80x9cNxe2x80x9d represents the number of measurements that are combined in the depth stacking.
A problem with depth stacking is that the stacking reduces the vertical resolution of the NMR measurements. Furthermore, the NMR tool that is used to obtain the measurements for the depth stacking may move between measurements. Thus, in thinly laminated sand-shale sequences, the measurements for sand and shale layers may be stacked together, thereby making it difficult to distinguish a shaley sand from a sequence of shales and highly producible sands. There are several techniques that may used to estimate the permeability of a formation, and these techniques may include fitting the NMR signal to a model function, a technique that may increase the statistical error in the derived permeability estimator. For example, one technique to derive a permeability estimator includes representing the amplitude of each spin echo signal by a summation, as described below:                                           echo            ⁡                          (              n              )                                ≈                                    ∑              j                        ⁢                                          A                j                            ⁢                              ⅇ                                                      -                    n                                    ⁢                                      TE                                          T                                              2                        ,                        j                                                                                                                                ,                            (                  Eq          .                      xe2x80x83                    ⁢          2                )            
where xe2x80x9cTExe2x80x9d represents the echo spacing, and xe2x80x9cAjxe2x80x9d represents the amplitude of components having a relaxation time T2,j. A histogram 17 of the Aj coefficients defines a T2 distribution, as depicted in FIG. 2. The Aj coefficients may be used in two different techniques to derive a permeability indicator, as described below.
In a technique referred to as the Timur-Coates technique, a bound fluid volume (BFV) cutoff time (called T2CUTOFF) is used. In this manner, the Aj coefficients for polarization times below the T2CUTOFF time may be summed to derive the BFV, as described by the following equation:                               BFV          =                                    ∑                              j                =                1                                            j                ⁢                                  xe2x80x83                                ⁢                max                                      ⁢                          A              j                                      ,                            (                  Eq          .                      xe2x80x83                    ⁢          3                )            
where xe2x80x9cjmaxxe2x80x9d corresponds to the T2 value of a cutoff time called T2CUTOFF. From the computed BFV, the Timur-Coates permeability (called KTC) may be estimated using the following equation:                                           K            TC                    =                                                    αφ                m                            ⁡                              (                                                      φ                    -                    BFV                                    BFV                                )                                      n                          ,                            (                  Eq          .                      xe2x80x83                    ⁢          4                )            
where a, m and n are adjustable parameters, and xe2x80x9cxcfx86xe2x80x9d represents a porosity that is obtained from analysis of the NMR data or from an independent measurement.
Another way to derive a permeability estimator using the histogram 17 is to compute a mean of the log (T2) times, often referred to as T2LM, that is described below by the following equation:                               log          ⁢                      xe2x80x83                    ⁢                      T2            LM                          =                              ∑                                          A                j                            ⁢              log              ⁢                              xe2x80x83                            ⁢                              T2                i                                                          ∑                          A              j                                                          (                  Eq          .                      xe2x80x83                    ⁢          5                )            
From the T2LM time, a permeability estimator may be derived as follows:
KSDR=axe2x80x2xcfx86mxe2x80x2(T2LM)Nxe2x80x2xe2x80x83xe2x80x83(Eq. 6)
where axe2x80x2, mxe2x80x2 and nxe2x80x2 are adjustable parameters.
A drawback of the above-described techniques is that once the NMR measurements are performed, several processing steps (such as the steps that are used to derive a distribution of relaxation times, for example) are used to derive the permeability estimator. Unfortunately, these processing steps may increase the statistical error of the derived permeability estimator.
It is also possible to derive a permeability estimate from NMR data without explicitly fitting the NMR signal. For example, U.S. Pat. No. 4,933,638, entitled xe2x80x9cBorehole Measurement of NMR Characteristics of Earth Formations, and Interpretation Thereof,xe2x80x9d granted Jun. 12, 1990, discloses the following technique to estimate a permeability. First, several magnetization levels (called M(tp1), M(tp2), . . . M(tpN)) of the MZ magnetization curve are measured using several polarization times (tp1, tp2, . . . tpN). Each M(tpi) magnetization level may be described by the following equation:                               M          ⁡                      (                          tp              i                        )                          =                              M            0                    (                                    (                              1                -                                  ⅇ                  ⁢                                                            -                                              tp                        i                                                                                    T                      1                                                                                  )                        ,                                              (                  Eq          .                      xe2x80x83                    ⁢          7                )            
where xe2x80x9cixe2x80x9d represents an integer from 1 to N. Next, the M(tpi) magnetization levels may be used to derive a piecewise linear graph that roughly approximates the MZ magnetization curve. The area (called A) under the piecewise linear graph may be calculated as described by the following equation:                     A        =                              ∑                          i              -              1                                      N              -              1                                ⁢                                    [                                                M                  ⁡                                      (                                          tp                      N                                        )                                                  -                                  M                  ⁡                                      (                                          tp                      i                                        )                                                              ]                        ·                          (                                                tp                                      i                    +                    1                                                  -                                  tp                  i                                            )                                                          (                  Eq          .                      xe2x80x83                    ⁢          8                )            
From the A area, a permeability (called K) may be calculated, using the following equation:
K=A2xcfx86tmxe2x88x922,xe2x80x83xe2x80x83(Eq. 9)
where xe2x80x9cxcfx86txe2x80x9d represents a porosity that is independently measured, and xe2x80x9cmxe2x80x9d represents an integer. However, this method employs T1 based measurements, which are relatively time-consuming and therefore impractical for the purposes of logging. Furthermore, Eq. 9 requires an independent measure of the porosity, "PHgr", which may not necessarily be available.
Thus, there is a continuing need for a technique that addresses one or more of the problems that are stated above.
In one embodiment of the invention, a method for use with spin echo signals that are received from a sample includes summing indications of the amplitudes of the spin echo signals. The results of the summing are used to determine an indication of a permeability of the sample, without using a distribution of relaxation times in the determination.
In another embodiment of the invention, a method for use with spin echo signals that are received from a sample includes summing products of indications of the amplitudes of the spin echo signals. The results of the summing are used to determine an indication of a permeability of the sample, without using a distribution of relaxation times in the determination.
The permeability indicator, derived by summing indications of echo amplitudes or products of echo amplitudes, may be used to provide a qualitative indication of formation quality to aid in establishing potential reserves.
Advantages and other features of the invention will become apparent from the following description, drawing and claims.